Saturday, March 31, 2007

There's a curious and bizarre fact about the universe that is introduced in physics courses without anyone stopping to point out just how curious and bizarre it is. I think it deserves to be more widely known to non-physicists. It's a little esoteric, but not so esoteric it can't be grounded in ordinary everyday concepts. And it isn't something hypothetical but something that is fundamental to modern physics and testable in the lab. It is also the reason why I chose to use theta/2 instead of theta in my quantum computing code.

This post will be in two parts. This part will mainly be about geometry and topology, and in the second part I'll talk about the physics.

In order to explain the idea I want to introduce the concept of a universal cover. If you look at the definition of this notion anywhere it can seem pretty daunting to non-mathematicians, but in actual fact the key idea is so simple that even a child could understand it. I can say that without fear of exaggeration because universal covers used to play a role in a childhood fantasy of mine. (OK, I was a weird kid, but not that weird.)

My crazy idea as a kid was this: if there were two ways to get from A to B, maybe the B you reached by taking either of the two routes were actually slightly different, even though they looked the same. I was actually more concerned with journeys from A to A. Suppose the kitchen has a door at each end and that you start in the bedroom, go down to the kitchen at one end, come out the other end, and return by a different path to the bedroom. How do you know this is the same bedroom? (I also admit I read a lot of science fiction as a kid.)

Suppose instead you took a simpler path: you left the bedroom and went to another room and then returned by the path you went. If you had taken a piece of string and tied one end to the bedroom and walked this path playing out the string behind you as you went then when you returned to your bedroom you would have both ends of the string visible. You could then reel in the string from your journey, without letting go of the ends, and see that indeed, both ends were in fact the ends of the same string and so you had retuned to the start of your journey. But if you'd taken the round trip through the double-doored kitchen you wouldn't be able to pull the string tight and so you could no longer be sure.

In what follows we consider two paths from A to B equivalent in this way: mark the two paths with strings going from A to B. If you can pull one string to line up exactly with the other (you're allowed to stretch) without breaking the string or moving the endpoints then they are considered equivalent. (The correct technical term is homotopic.) So if a path that appears to go from A to A is homotopic to the trivial zero length path, then you know that the path really is from A back to A. But if you can't shrink the path down to zero length you're faced with the possibility that the string is extended from A to a similar looking place, A'. (Of course this implies a considerable conspiracy with someone tying a distinct identical string to an identical bedroom, but if you can't see both ends, alien beings from a higher dimension could do anything...)

The best case scenario is that all of the bedrooms are the same - a conventional house. Call this house X. But what's the worst case scenario? How many different copies of the bedroom might there be? Well let's call the starting bedroom 0. If we walk anti-clockwise round the house, going through the kitchen, we end up back at a bedroom. In the worst case scenario this will be another bedroom. Let's call it 1. Walk round again and we get to another copy of the bedroom, call it 2 and so on. So we have an infinite sequence of bedrooms, (0,1,2,…). But we could have walked clockwise. In that case we'd get a sequence (-1,-2,…). So now think about the set of all points in this infinite house. Each point can be described as a pair (x,n) where x is a point in the house X and n labels which of the infinite number of paths you could have taken to get there. (Remember, if the two paths are homotopic we consider them to be the same path.)

If using a house as an example seems too far-fetched think of a mult-storey car park. It's easy to walk what you think is a closed loop but find that you've in fact walked to almost identical looking, but different, floor.

(Incidentally, this kind of infinite house is the kind of thing that can be created in a Portal engine such as in the game Portal.)

Let's think about this more generally. Given a topological space X we can form another space X' from it by (1) picking a point p in it and (2) forming the set of all pairs (x,n) where x is a point in X and n is a path (up to homotopy) from p to x. X' is known as the universal cover of X.

Enough with houses, let's try a more purely geometric example. Consider the space of rotations in 2D, otherwise known as SO(2). A rotation can simply be described by an angle with the proviso that a rotation through an angle θ is the same as a rotation through θ+2nπ where n is an integer. The space of rotations is essentially the same thing as the circle (by circle I mean the 1-dimensional boundary of a 2-dimensional disk). Looping once round the circle brings you back to where you started. So how can we visualise the universal cover of SO(2)? There's a nice trick for this. Imagine a disk that's free to rotate around one axis. SO(2) describes the rotations of this disk. But now imagine a belt, fixed at one end, connected at the other end to the centre of this disk like this:

The rule is that we're allowed to twist the belt as much as we like as long as the centre line of the belt is along the axis. Notice how if we rotate the disk through an angle of 2nπ the disk returns to its starting point, but the belt 'remembers' how many twists we took to get there. Notice another thing: if we travel along the belt from one end to the other, the orientation of the belt at each point gives a 2D rotation. In other words, the belt itself represents a path in SO(2) with one one representing a fixed point in SO(2) and the other end matching the current orientation of the disk. So if the orientations of the disk represent elements of SO(2), the possible twistings of the belt represent elements in the universal cover of SO(2). And what the twists do is allow us to differentiate between rotations through θ and rotations through θ+2nπ. In other words, the universal cover is just the real line, but not folded down so that θ is the same as θ+2nπ.

The universal cover is the worst case scenario where all non-homotopic paths are considered to represent distinct points. But it's also possible to construct an in-between scenario where some paths that aren't homotopic to each other are still considered to lead to the same place. For example, take the real line again. If we identify 0 and 2π we get SO(2). If we consider 0 and 2nπ to be distinct for n≠0 we get the universal cover. But we pick an N≠1 and consider 2Nπ to be the same as 0, but that 2nπ is distinct from 0 for 0<n<N. What we now end up with is an N-fold cover of SO(2). Unfortunately I don't know a way construct a physical example of a general N-fold cover using belts and discs. (At least for N>2. For N=2 you should easily be able to come up with a mechanism based on what's below.)

With the examples so far, the universal cover of a space has always had an infinite number of copies of the original space. That isn't always the case and now we'll meet an example where the universal cover is a double cover: the real projective plane, RP2. The easiest way to visualise this is to imagine the game of Asteroids played on a circular TV screen. In the usual Asteroids, when you go off one side of the screen you reappear on the opposite side. In this version it's much the same, points on the edge that are diametrically opposite are considered to be the same points. Such a universe is a real projective plane. Let P be the point at the centre of the screen. Suppose we attach one end of our string to P and fly off towards the edge, reappearing on the other side. Eventually we get back to P again. Can we shrink this path down to a point? Unfortunately, any path that crosses the edge of our screen will always cross the edge no matter how much we try to deform it. If we move the point at which it crosses one edge of the screen, the point at the other side always remains diametrically opposite.

This means you can never bring these points together in order to collapse down a loop. On the other hand, suppose we loop twice round the edge. The following diagram shows how this path can be shrunk down to a loop:

So given two distinct points in RP2 there are two inequivalent paths between them, and so the universal cover of RP2 is made from two copies of the space and is in fact a double cover.

Now we'll try a 3D space - the space of 3D rotations, otherwise known as SO(3). This space is 3-dimensional because it takes 3 numbers to define a rotation, for example we can specify a rotation using a vector whose direction is the axis of rotation and whose length is the angle of rotation. Now all possible rotations are rotations through an angle 0≤θ≤π for some exis. So every rotation can be represented as a point in the ball of radius π. But a rotation around the axis a through an angle of π is also the same as a rotation around the axis -a through π. In other words, SO(3) is the closed 3-dimensional ball with diametrically opposite points identified. This is very similar to the situation with RP2. In fact, almost exactly the same argument shows that the universal cover of SO(3) is the double cover. (For an alternative wording, see this posting.) There is also another way of seeing this that is analogous to the disk with belt example above. This time we have a sphere instead of a disk. The belt is fixed at one end and the other end is attached to the sphere. We are free to rotate the sphere however we want, and we're no longer confined to having the belt along one axis. It is an amazing but easily demonstrable fact that if you rotate the sphere through 2π, you cannot untwist the belt, but if you rotate it through 4π you can! This is known as the Dirac Belt Trick, and if you don't believe it, watch the Java applet written by science fiction author Greg Egan.

If you've done much computer graphics you'll have already met the double cover of SO(3). Computer graphics people sometimes like to represent rotations using unit quaternions. The unit quaternions form a 3-dimensional sphere but there are two such quaternions for each rotation. So the unit quaternions actually represent this double cover, and SO(3) is a 3-dimensional sphere with antipodean points identified.

And that's the geometry dealt with. In the next installment I'll say what this has to do with electrons and my quantum computing code.

Footnote

Balls and Spheres. The closed n-dimensional ball is the set of points (x1,...,xn) such that x12+...+xn2≤1. The (n-1)-dimensional sphere is the set of points (x1,...,xn) such that x12+...+xn2=1.

5 comments:

Derek Elkins
said...

Another way to demonstrate the double cover that you can do in the comfort of your own home is to rotate your hand at the wrist keeping your palm facing upward at all times through 360 and then 720 degree rotation. At 360 degrees your arm certainly will not be in it's original state, but it will at 720.

How about, for an N-fold cover, take a belt that's N times the circumference of a wheel, and span it out with another wheel, mounted just to help keeping things tight - resulting in something vaguely like a bicycle chain mechanism.

Now, for each revolution of the wheel, viz every 2π rotation, you end up at a different point along the belt, but after N of them, you get back to the point you started.

My quantum mechanics prof demonstrated SU(2) using a belt as well. I thought he was just a little eccentric, I didn't realise it was a trick due to Dirac.

There was a talk at Perimeter Institute a while ago by Sundance Bilson-Thompson, who is trying to build a preon model out of braids and ribbons. If his model is true, the electron is described by SU(2) because it is really made of tiny belts!

Possibly worth pointing out that the universal cover is actually "universal" in the technical sense that it's initial in some appropriate category, here the category of covers of X and homotopy classes of (surjective?) maps.